c) After the perturbation is added, the Hamiltonian can be solved
exactly by completing the square on the
where the displacement The new ground state energy and
eigenfunction are
The harmonic oscillator vibrates about the new equilibriumpoint with
the same frequency as before. The constants and are unchanged by
d) To find the probability that a particle, initially in the ground state,
remains in the ground state after switching on the potential, we employ the
sudden approximation. Here we just evaluate the overlap integral of the
two eigenfunctions, and the probability is the square of this overlap:
5.21 Cut the Spring! (MIT)
a)Below wegive the Hamiltonian the frequency and the eigenvalues
of the particle while coupled to two springs:QUANTUM MECHANICS 265